2-Domination number of generalized Petersen graphs

  • Davood Bakhshesh
  • Mohammad Farshi
  • Mohammad Reza Hooshmandasl
Article
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Abstract

Let \(G=(V,E)\) be a graph. A subset \(S\subseteq V\) is a k-dominating set of G if each vertex in \(V-S\) is adjacent to at least k vertices in S. The k-domination number of G is the cardinality of the smallest k-dominating set of G. In this paper, we shall prove that the 2-domination number of generalized Petersen graphs \(P(5k+1, 2)\) and \(P(5k+2, 2)\), for \(k>0\), is \(4k+2\) and \(4k+3\), respectively. This proves two conjectures due to Cheng (Ph.D. thesis, National Chiao Tung University, 2013). Moreover, we determine the exact 2-domination number of generalized Petersen graphs P(2kk) and \(P(5k+4,3)\). Furthermore, we give a good lower and upper bounds on the 2-domination number of generalized Petersen graphs \(P(5k+1, 3), P(5k+2,3)\) and \(P(5k+3, 3).\)

Keywords

Generalized Petersen graph k-domination number \(\alpha \)-domination number 

2010 Mathematics Subject Classification

05C69 

Notes

Acknowledgements

The authors would like to thank the reviewers for their comments to improve the paper and clarify some of the proofs.

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Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  • Davood Bakhshesh
    • 1
  • Mohammad Farshi
    • 1
  • Mohammad Reza Hooshmandasl
    • 1
  1. 1.Department of Computer Science, Combinatorial and Geometric Algorithms LaboratoryYazd UniversityYazdIran

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